Hyperelliptic odd coverings

We investigate a class of odd (ramification) coverings C → ℙ1 where is hyperelliptic, its Weierstrass points map to one fixed point and the covering makes hyperelliptic involution commute with an ℙ1. show that total number minimal degree 4g \(\left({\matrix{{3g} \cr {g - 1} \cr}} \right){2^{2g}}\) when general. Our study approached from three main perspectives: if effective theta characteristic they are described as solution certain differential equations; then studied monodromy viewpoint deformation argument leads final computation.